Theorem scafvalg 14616

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	|- B = ( Base ` W )
scaffval.f	|- F = (Scalar ` W )
scaffval.k	|- K = ( Base ` F )
scaffval.a	|- .xb = ( .sf ` W )
scaffval.s	|- .x. = ( .s ` W )

Assertion

Ref	Expression
scafvalg	|- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Proof of Theorem scafvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 |- B = ( Base ` W )
2		scaffval.f	. . . 4 |- F = (Scalar ` W )
3		scaffval.k	. . . 4 |- K = ( Base ` F )
4		scaffval.a	. . . 4 |- .xb = ( .sf ` W )
5		scaffval.s	. . . 4 |- .x. = ( .s ` W )
6	1, 2, 3, 4, 5	scaffvalg 14615	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
7	6	3ad2ant1 1049	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
8		oveq12 6084	. . 3 |- ( ( x = X /\ y = Y ) -> ( x .x. y ) = ( X .x. Y ) )
9	8	adantl 277	. 2 |- ( ( ( W e. V /\ X e. K /\ Y e. B ) /\ ( x = X /\ y = Y ) ) -> ( x .x. y ) = ( X .x. Y ) )
10		simp2 1029	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> X e. K )
11		simp3 1030	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> Y e. B )
12		vscaslid 13494	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
13	12	slotex 13357	. . . . 5 |- ( W e. V -> ( .s ` W ) e. _V )
14	5, 13	eqeltrid 2325	. . . 4 |- ( W e. V -> .x. e. _V )
15	14	3ad2ant1 1049	. . 3 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .x. e. _V )
16		ovexg 6109	. . 3 |- ( ( X e. K /\ .x. e. _V /\ Y e. B ) -> ( X .x. Y ) e. _V )
17	10, 15, 11, 16	syl3anc 1278	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .x. Y ) e. _V )
18	7, 9, 10, 11, 17	ovmpod 6206	1 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   Basecbs 13330  Scalarcsca 13411   .scvsca 13412   .sfcscaf 14597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-ndx 13333  df-slot 13334  df-base 13336  df-sca 13424  df-vsca 13425  df-scaf 14599

This theorem is referenced by:  lmodfopne  14635